where \(E\)  denotes a resolution of the identity associated with \(L\) ; see e.g. [1]}. When the spectrum is discrete
\(\sigma (L)=\lbrace {\lambda }_n\rbrace _{n=0}^\infty \) , and \(\lbrace {\varphi }_n\rbrace _{n=0}^\infty \)  is an orthonormal basis of eigenfunctions, then (REF ) takes the form
\(L^\sigma f\,=\,\sum _{n=0}^\infty \,{\lambda }_n^\sigma \,\langle f,{\varphi }_n\rangle \,{\varphi }_n,\) 
